May 26, 1978 - under Electronic Systems Division Contract F19628-78-C-0002 by ..... utilized for all machine examples. On the Illiac, two .... STAR PROGRAM BREAKDOWN .... nevertheless provide competitive performance at very low cost. ..... CDC7600 - $3M, IBM370-168 - $2M, CDC6600 - $1M, PDP11-70 - $150K.
- I H-78-98
Technical Note
1978-16 R. S. Bucy K. D. Senne
New Frontiers in Nonlinear Filtering 26 May 1978 Prepared for the Department of the Air Force under Electronic Systems Division Contract F19628-78-C-0002 by
Lincoln Laboratory MASSACHUSETTS INSTITUTE OF TECHNOLOGY LEXINGTON, MASSACHUSETTS
Approved for public release; distribution unlimited.
\)
O
At)Aos R
and h: Rn
— RS
s=2 and n=3 for the three-dimensional model or n=2 for the two-dimensional
A
= -ßx
3 dx.
+x
2 di
+
2%3
+
2^22
3x'
H(x3) cos (x1) h
=
R
=
(2.4)
H(x3) sin (xx) / z 1\ z
= \;
Equation (2.3) is the Stratonovich-Kushner equation for the solution to the nonlinear filtering problem (see [4]). Two alternatives exist for solving (2.3) on a digital computer [2], One scheme involves the direct replacement of (2.3) with a suitable finite difference equation which in the continuous limit approaches (2.3), and when solved yields a solution which also in the limit (hopefully) approaches the solution of (2.3).
A more effective method, however, is to pose and solve
a sequence of discrete filtering problems as z
n
=
H(x ) cos (x ) n n
+ V
n (2.5)
2
z
n
n 2 n x
n
3
1
=
H(x ) sin (x ) n n
=
x n-1
+
=
2 x n-1
+
=
x
3
+
A x
n-1
W OA
(2.6)
n-1 3
.. + 3Ax . n-1 n-1
+
w
3
. n-1
2 where E(vX) = r/A , E(w1) Aq . with A interpreted as the sampling n ' n interval. The solution to the discrete filtering problem evolves according to the equations [4], [8]
P ,-, n+1
=
S*F
(2.7)
n
F=^D«P=-^D-S*F1 n K n n K n n-1 n n
(2.8)
12 3 i {F } is the conditional distribution of x , x , and x given z ., , nn... . nn n° n-1 z 0 , ..., z {z ,z .,..., z }, * denotes convolution, • denotes pointn-2 o n n-1' o ' wise multiplication. The functions S and D are derived from (2.5) and (2.6), where P
based on assumed probability density functions (Gaussian) for w respectively, and K r J
n
is a scalar which is chosen to normalize F
and v n
,
to have unit
total mass. For the two-dimensional special case (i.e., without the amplitude signal process) the optimal filter recursion of (2.8) becomes
F
" r W j
F
°**\ li^K-
n-1 CyrVA.u)dp
(2.9)
where Dn(yi)=
1 2 z cos (y-) 4- z sin (y. J J ) n r n l exp^ ^-^ ^->
(2.10)
It can be shown [32], [35] that a cyclically modulated density F , defined as
F (y
n
with -7T _< y
l'y2} " /
)
F
n
(y
l
+ 2TTk
'
y
2
+ 2TT£/A)
(2,11)
< 7T, and -TT/A £ y? < TT/A, carries all of the information necessary 1 for nonlinear filtering subject to the cyclic loss function L(£..) = y(l-cos(e1)),
where e
is the error in the estimate of the phase x .
The modulated density
satisfies the recursion relation TT/A
V0'T) ■ rn Va)
(2.12)
S(T-y)F(a-uA,y)du -TT/A
where
r
S(T-y) =
exp
I
(T-y 4- 2-TTi/A)' 2q22A
(2.13)
1=-°°
It has also been established [32], [35] that the estimate x
n
of x
n
which
minimizes the cyclic loss function is given by
E e
xn = arg
1 n
1
z . , z 2.
l
(2.14)
i < n) —
\
The optimal filter recursion (2.8) for the three-dimensional problem can be obtained in an analogous fashion.
Using a discrete form
of the representation theorem (see [4]) and following the development of the two dimensional special case, the density F
for the filtered amplitude
and cyclic phase processes can be updated recursively as
°°
TT/A
Fn(a,i,a) = |- Dn(a,a) I -°°
F
n-l
S1(x-a)S2(a-3n). -TT/A
(a-uA, y,n)dydn
(2.15)
where
D (a,a) = exp [ — H(a) (z
cos a + z
sin a - y H(a))]
oo
51(T-a) = y
exp [- ^— (i-a + -7- ) ] 2q AT 22
k=-°°
S'2-0(a-ßn) = and K
1 2 exp [- -r-—T(a-Bn) ] 2
